Delaunay graphs are almost as good as complete graphs

Let S be any set of N points in the plane and let DT(S) be the graph of the Delaunay triangulation of S. For all points a and b of S, let d(a, b) be the Euclidean distance from a to b and let DT(a, b) be the length of the shortest path in DT(S) from a to b. We show that there is a constant c(≤ 1+√5/2 π ≈ 5.08) independent of S and N such that DT(a, b)/d(a, b) ≪ c.